Modeling an elastic stiffness tensor in a transverse isotropic subsurface medium

ABSTRACT

Modeling an elastic stiffness tensor in a transverse isotropic subsurface medium acquires well log data for at least one well passing through the transverse isotropic subsurface medium. The transverse isotropic subsurface medium is divided into an effective anisotropic layer and an isotropic layer. The effective anisotropic layer elastic parameters are modeled, and the isotropic layer elastic parameters are modeled using the effective anisotropic layer elastic parameters and the acquired well log data. The modeled effective anisotropic layer elastic parameters and the modeled isotropic layer elastic parameters are used to upscale the effective anisotropic layer and the isotropic layer into the transverse isotropic subsurface medium comprising a single layer and to determine the five members of the elastic stiffness tensor for the transverse isotropic subsurface medium.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority and benefit from U.S. ProvisionalPatent Application No. 62/084,041, filed Nov. 25, 2014, for “ModellingElastic Stiffness Tensor in a TI Medium”, the entire contents of whichis incorporated herein by reference.

TECHNICAL FIELD

Embodiments of the subject matter disclosed herein generally relate tomethods and systems for imaging and characterizing a subsurface.

BACKGROUND

Velocity anisotropy, which is known as the directional dependency ofvelocities, is important in subsurface imaging and characterization.Most elasticity theories consider an isotropic medium as their mainassumption for addressing the problems in the field of reservoirgeophysics. This assumption is challenged by the reality of thesubsurface, which could be made up of structures such as beds andfractures and which has gone through a complex geological history. Thesefactors can make the subsurface of the Earth deviate significantly fromthe isotropic assumption used in the routine algorithms and approaches.

In general, four classes of anisotropy are defined that range from acompletely isotropic medium (with two elastic constants) to a completelyanisotropic medium (with 21 elastic constants). The four classes referto specific conditions where the number of elastic stiffness constantscan be reduced. The four classes are named Cubic with 3 independentelastic constants, Transverse Isotropic (TI) with 5 independent elasticconstants, Orthorhombic with 9 independent elastic constants andMonoclinic with 13 independent elastic constants. A TI medium providesthe closest description of sedimentary rock.

Conventionally, anisotropy in the context of isotropic approaches ishandled using Thomsen parameters and approximation. Thomsen suggestedthree parameters to correct for anisotropy effects in weak-anisotropymediums. These parameters, ϵ, δ and γ, are now used regularly in allreservoir geophysics disciplines to address anisotropy effects. However,calculation of the Thomsen parameters requires information such aslaboratory data or well tracks in different directions compared with thesymmetry axis which are expensive to apply in practice. Therefore, theneed still exists to improve cheaper methods for calculating anisotropyparameter in the form of stiffness tensor or Thomsen parameters in a TIsubsurface.

SUMMARY

Embodiments are directed to systems and methods that utilize a rockphysics workflow to model the elastic stiffness tensor in a transverseisotropic (TI) medium using conventional well-log suites, i.e.,conventional suites of well-log data. The workflow uses downscalingfollowed by upscaling of normal logs by the Backus model. The rockphysics modelling is performed within the downscaling step. Theresulting downscaling step is where anisotropy information is integratedwith the well log information through the rock physics models. Theseanisotropy factors are in a scale much smaller than the wavelength, andthe anisotropic layer is seen as an effective isotropic medium.Following downscaling with rock physics modelling using the integratedanisotropy information, the modelled logs are upscaled to the measuredones. This workflow provides a first estimate of anisotropic effects andcan be seen in the context of boundary models to define the possibleanisotropy boundary of an area. Therefore, the workflow assists in thedetermination of possible ranges of anisotropy changes in a verticalwell within a TI medium. Furthermore, the workflow can be used todetermine the Thomsen parameters, i.e., ϵ, δ and γ, ranges when no otherdata is available except conventional well-logs. The results of thismodel can be improved through integration with seismic (e.g. inversion)or laboratory (e.g. core analysis) data.

Embodiments are directed to a method for modeling an elastic stiffnesstensor in a transverse isotropic subsurface medium. Well log data areacquired for at least one well passing through the transverse isotropicsubsurface medium. In one embodiment, the well log data are acquired ina direction parallel to a symmetry axis passing through layers in thetransverse isotropic subsurface medium.

The transverse isotropic subsurface medium is divided into ananisotropic layer and an isotropic layer such that a sum of ananisotropic layer volume fraction and an isotropic layer volume fractionequals one. Effective anisotropic layer elastic parameters are modelled.The effective anisotropic layer elastic parameters include ananisotropic layer density, an anisotropic layer p-wave velocity and ananisotropic layer s-wave velocity. In one embodiment, modeling theeffective anisotropic layer elastic parameters further includescalculating an anisotropic layer density as a volume weighted average ofall anisotropic component densities in the anisotropic layer andmodeling an anisotropic layer p-wave velocity and an anisotropic layers-wave velocity along a symmetry axis of the transverse isotropicsubsurface medium using a rock physics model selected in accordance witha source of anisotropy in the anisotropic layer. In one embodiment, therock physics model includes intrinsic factors in the anisotropy layer orextrinsic factors in the anisotropy layer. Modeling the effectiveanisotropic layer p-wave velocity and anisotropic layer s-wave velocityalso includes using additional anisotropy data containing at least oneof core data, core ultrasonic measurements for a plurality of wavepropagation angles and seismic data.

The isotropic layer elastic parameters are modeled using the anisotropiclayer elastic parameters and the acquired well log data. The isotropiclayer elastic parameters include an isotropic layer density, anisotropic layer p-wave velocity and an isotropic layer s-wave velocity.In one embodiment, modeling the isotropic layer elastic parametersincludes using measured p-wave velocity and measured s-wave velocityfrom the acquired well log data and modeled effective anisotropic layerp-wave velocity and effective anisotropic layer s-wave velocity in asimplified Backus model for a two layer transverse isotropic medium andwave propagation normal to layering in the two layer isotropic medium tomodel the isotropic layer elastic parameters.

The modeled effective anisotropic layer elastic parameters and themodeled isotropic layer elastic parameters are used to upscale theeffective anisotropic layer and the isotropic layer into the transverseisotropic subsurface medium having a single layer and to determine theelastic stiffness tensor for the transverse isotropic subsurface medium.

In one embodiment, using the modeled effective anisotropic layer elasticparameters and the modeled isotropic layer elastic parameters to upscalethe effective anisotropic layer and the isotropic layer into thetransverse isotropic subsurface medium includes using the effectiveanisotropic layer elastic parameters to determine two effective Lame'sparameters (λ_(A) and μ_(A)) and using the isotropic layer elasticparameters to determine two Lame's parameters (λ_(I) and μ_(I)). Inaddition, the effective anisotropic layer Lame's parameters (λ_(A) andμ_(A)) and the isotropic layer Lame's parameters (λ_(I) and μ_(I)) arecombined to yield five independent members of the transverse isotropicsubsurface medium elastic tensor (C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄). In oneembodiment, a full Backus model is used. In one embodiment, the fiveindependent transverse isotopic subsurface elastic tensor members (C₁₁,C₁₂, C₁₃, C₃₃ and C₄₄) are used to calculate transverse isotropicsubsurface medium p-wave velocities and transverse isotropic subsurfacemedium s-wave velocities for a plurality of wave propagation angles withrespect to an axis of symmetry in the transverse isotropic subsurfacemedium.

In one embodiment, using the modeled effective anisotropic layer elasticparameters and the modeled isotropic layer elastic parameters to upscalethe effective anisotropic layer and the isotropic layer into thetransverse isotropic subsurface medium includes using the effectiveanisotropic layer elastic parameters to determine effective anisotropiclayer Lame's parameters (λ_(A) and μ_(A)), using the isotropic layerelastic parameters to determine two Lame's parameters for the isotropiclayer (λ_(I) and μ_(I)), and combining the effective Lame's parametersfor the anisotropic layer and the Lame's parameters for the isotropiclayer to yield five independent transverse isotropic subsurface mediumelastic tensor members (C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄) for the subsurfacemedium.

Embodiments are also directed to a computer-readable medium containingcomputer-executable code that when read by a computer causes thecomputer to perform a method for modeling an elastic stiffness tensor ina transverse isotropic subsurface medium by acquiring well log data forat least one well passing through the transverse isotropic subsurfacemedium, dividing the transverse isotropic subsurface medium into ananisotropic layer and an isotropic layer such that a sum of ananisotropic layer volume fraction and an isotropic layer volume fractionequals one, modeling effective anisotropic layer elastic parameters,modeling isotropic layer elastic parameters using the anisotropic layerelastic parameters and the acquired well log data and using the modeledeffective anisotropic layer elastic parameters and the modeled isotropiclayer elastic parameters to upscale the effective anisotropic layer andthe isotropic layer into the transverse isotropic subsurface mediumcomprising a single layer and to determine the elastic stiffness tensorfor the transverse isotropic subsurface medium.

Embodiments are also directed to a computing system for modeling anelastic stiffness tensor in a transverse isotropic subsurface medium.The computing system includes a storage device containing well log datafor at least one well passing through the transverse isotropicsubsurface medium and a processer in communication with the storagedevice. The processor is configured to divide the transverse isotropicsubsurface medium into an effective anisotropic layer and an isotropiclayer such that a sum of an effective anisotropic layer volume fractionand an isotropic layer volume fraction equals one, model effectiveanisotropic layer elastic parameters, model isotropic layer elasticparameters using the effective anisotropic layer elastic parameters andthe acquired well log data and use the modeled effective anisotropiclayer elastic parameters and the modeled isotropic layer elasticparameters to upscale the effective anisotropic layer and the isotropiclayer into the transverse isotropic subsurface medium comprising asingle layer and to determine the five independent members of theelastic stiffness tensor for the transverse isotropic subsurface medium(C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄).

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate one or more embodiments and,together with the description, explain these embodiments. In thedrawings:

FIG. 1 is a representation of an embodiment of stiffness tensor andvelocities for an isotropic medium;

FIG. 2 is a representation of an embodiment of stiffness tensor forisotropic and transverse isotropic mediums;

FIG. 3 is a schematic representation of an embodiment of a workflow formodeling an elastic stiffness tensor in a transverse isotropicsubsurface medium;

FIG. 4 is a flowchart of an embodiment of a method for modeling anelastic stiffness tensor in a transverse isotropic subsurface medium;and

FIG. 5 is a schematic representation of an embodiment of a computingsystem for use in executing a method for modeling an elastic stiffnesstensor in a transverse isotropic subsurface medium.

DETAILED DESCRIPTION

The following description of the embodiments refers to the accompanyingdrawings. The same reference numbers in different drawings identify thesame or similar elements. The following detailed description does notlimit the invention. Instead, the scope of the invention is defined bythe appended claims. Some of the following embodiments are discussed,for simplicity, with regard to local activity taking place within thearea of a seismic survey. However, the embodiments to be discussed nextare not limited to this configuration, but may be extended to otherarrangements that include regional activity, conventional seismicsurveys, etc.

Reference throughout the specification to “one embodiment” or “anembodiment” means that a particular feature, structure or characteristicdescribed in connection with an embodiment is included in at least oneembodiment of the subject matter disclosed. Thus, the appearance of thephrases “in one embodiment” or “in an embodiment” in various placesthroughout the specification is not necessarily referring to the sameembodiment. Further, the particular features, structures orcharacteristics may be combined in any suitable manner in one or moreembodiments.

Embodiments of systems and methods use a rock physics workflow to modelchanges in the elastic stiffness tensor due to anisotropy in a verticalwell, i.e., a well extending parallel to the symmetry axis. This elasticstiffness tensor can be used to calculate the Thomsen parameters or evento model velocities directly. Conventional well logs for the verticalwell and other wells passing through the subsurface are used as input.These well logs provide elastic parameters in the symmetry directionwithin a transverse-isotropic medium. In one embodiment, the informationobtained from the conventional well logs is tied with other anisotropyinformation to yield an estimation of anisotropy with increasedaccuracy.

Hooke's law for a general linear and elastic anisotropic solid gives asimple relationship to relate stress variations (σ_(ij)) and strainchanges (ϵ_(ij)) through a fourth-rank tensor referred to as thestiffness tensor (C_(ijkl) ) This stiffness tensor characterizes theelasticity of the medium using a total of 81 components and follows thelaws of tensor transformation. The symmetry of stresses and strains aswell as symmetry within the tensor itself reduces the number ofcomponents to 21 independent constants, which is the maximum number ofindependent elastic constants that any homogeneous linear elasticanisotropic medium can have.

However, the more common form Hook's law is the one using the Voigtnotation, which is summarized as:

σ_(I)=C_(ij) ϵ_(J)  (1)

This form is more popular due to its simplicity in calculations byreducing the indices. Here, C_(ij) matrix has 21 independent elasticconstants for an anisotropic medium (same as the C_(ijkl) tensor). Thecomponent number of stiffness elastic matrix reduces to two (Lame'sparameters λ and μ) for an isotropic linear elastic material. FIG. 1shows a simple sketch that expresses C_(ij) matrix and its componentsfor an isotropic medium along with isotropic velocities (V_(p) andV_(s)) which are can be expressed based on these stiffness constants.

Referring to FIG. 1, an isotropic medium 102, for example, an isotropicsubsurface medium has a stiffness tensor 104 illustrated as a matrixcontaining a plurality of elastic stiffness components. The isotopicmedium also has associated velocities 106 for waves propagating throughthe medium, including a p-wave velocity 108 and an s-wave velocity 110.However, the number of elastic stiffness components within the stiffnesstensor C_(ij) matrix increases as the medium approaches a state ofanisotropy. Therefore, the equations described above for an isotropicmedium will no longer be valid as the number of elastic stiffnessconstants increases, and the isotropic velocities are not sufficientlyaccurate for an anisotropic medium. The accuracy of the isotropicvelocities decrease as the subsurface medium becomes more anisotropicdue to an increase in the number of elastic constants.

Regarding anisotropy in TI mediums and Thomsen parameters, TI mediumsare defined as materials that show isotropy in one direction andanisotropy in a direction perpendicular to the isotropy plane. Thedirection of anisotropy is normally referred to as the symmetry axis. Insuch conditions, the number of C_(ij) matrix constants increases to fiveindependent components, and isotropic velocities are changed accordinglyas below:

$\begin{matrix}{{V_{QP} = \sqrt{\frac{{c_{11}\sin^{2}\theta} + {c_{33}\cos^{2}\theta} + c_{44} + \sqrt{M}}{2\rho}}}{V_{QSV} = \sqrt{\frac{{c_{11}\sin^{2}\theta} + {c_{33}\cos^{2}\theta} + c_{44} - \sqrt{M}}{2\rho}}}{V_{SH} = \sqrt{\frac{{c_{66}\sin^{2}\theta} + {c_{44}\cos^{2}\theta}}{\rho}}}} & (2) \\{M = {\left\lbrack {{\left( {c_{11} - c_{44}} \right)\sin^{2}\theta} - {\left( {c_{33} - c_{44}} \right)\cos^{2}\theta}} \right\rbrack^{2} + {\left( {c_{13} + c_{44}} \right)^{2}\sin^{2}2\theta}}} & \;\end{matrix}$

Here, V_(QP) and V_(QSV) are the quasi-longitudinal mode and quasi-shearmode velocities, while V_(SH) is the horizontal shear velocity. θ is theangle between the wave vector and the symmetry axis of the TI medium.For a TI subsurface medium, the symmetry axis is perpendicular to theisotropic layers in the TI subsurface medium. For wave vectorspropagating along or parallel to the symmetry axis, θ=0.

Thomsen (1986) approximated these velocities in a TI medium for a weaklyelastic anisotropic scenario as follows:

$\begin{matrix}{{V_{QP} \approx {{V_{P}(0)}\left( {1 + {{\delta sin}^{2}{\theta cos}^{2}\theta} + {ɛ\; \sin^{4}\theta}} \right)}}{V_{QSV} \approx {{V_{S}(0)}\left\lbrack {1 + {\frac{V_{P}(0)}{V_{S}(0)}\left( {ɛ - \delta} \right)\sin^{2}{\theta cos}^{2}\theta}} \right\rbrack}}{V_{SH} \approx {{V_{S}(0)}\left( {1 + {{\gamma sin}^{2}\theta}} \right)}}} & (3)\end{matrix}$

Here, ϵ, δ and γ, which are known as Thomsen anisotropic parameters, aredefined as:

$\begin{matrix}{{ɛ = \frac{c_{11} - c_{33}}{2c_{33}}}{\delta = \frac{\left( {c_{13} + c_{44}} \right)^{2} - \left( {c_{33} - c_{44}} \right)^{2}}{2{c_{33}\left( {C_{33} - C_{44}} \right)}}}{\gamma = \frac{c_{66} - c_{44}}{2c_{44}}}} & (4)\end{matrix}$

and V_(P)(0) and V_(S)(0) are the velocities along the symmetry axiswith following definitions:

$\begin{matrix}{{V_{P}(0)} = \sqrt{\frac{c_{33}}{\rho}}} & (5) \\{{V_{S}(0)} = \sqrt{\frac{c_{44}}{\rho}}} & \;\end{matrix}$

These two velocities, which represent the slowest velocities in the TImedium, can also be calculated using equation (2) with θ=0. In otherwords, the Thomsen equation calculates velocities in the symmetrydirection and then adds anisotropy effects into these velocities usingthe three Thomsen parameters. The result is that five independentelastic parameters having no physical meaning are translated into threeparameters (ϵ, δ and γ) with a certain physical meaning. In particular,ϵ, which is also referred to as P-wave anisotropy, represents thefractional difference between the P-wave velocities in the horizontaland vertical directions, and δ can be related to both near verticalP-wave velocity and angular SV-wave velocity variations. The parameter γhas the same role as ϵ but for S-wave velocity by representing thefractional difference between the SH-wave velocities in the horizontaland vertical directions. Therefore, these three parameters can be usedto describe a TI medium, and a measure of these parameters is used inthe anisotropic modeling workflow.

Following Thomsen, work on these three parameters and their combination(e.g. Alkhalifah and Tsvankin, 1995) modeled wave propagation in ananisotropic medium. However, the Thomsen approximation is valid for weakanisotropy, and calculating ϵ, δ and γ is not easy. In the following,the anisotropy problem in TI mediums is reviewed by looking atanisotropy effects on the stiffness matrix C_(ij) using the Backus(1962) model. A workflow that only needs conventional well-logs is usedto model anisotropy in TI mediums. In one embodiment, this approach iscoupled to other anisotropic workflows for a more accurate anisotropicmodeling.

Regarding the Backus model and stiffness tensor in a VTI medium, Postma(1955) showed that in a heterogeneous media, anisotropy is ascale-dependent property where a two-layer layered medium can behave asan anisotropic medium if each of the two layers is isotropic in a finerscale than the wavelength of the seismic waves. Furthermore, Backus(1962) extended the work of Postma (1955) into general media with threeor more layers. Backus (1962) showed that in the long wavelength limit astratified medium composed of isotropic layers still can make a TImedium. This theory allows a bunch of layers to be replaced by a singleanisotropic layer or a single anisotropic medium to be decomposed into abunch of isotropic layers. This infers that anisotropy is a frequency(scale) dependent phenomenon, and fine isotropic layering (higherfrequency) can express itself as anisotropy on a larger scale (lowerfrequencies).

Conventionally, such a stratified medium with fine layering is replacedwith a homogenous, transversely isotropic material. In particular,anisotropy can be defined on different scales from large scale, e.g.,medium layering or fractures, down to fine scale, e.g., grain alignmentsor cracks, and the measurement scale decides if that medium behaves asisotropic or anisotropic. For normal incidence seismic wave propagation,when these anisotropy features (like the stratified medium) are on ascale much finer than the wavelength of seismic waves, the waves willaverage their elastic properties, and the medium will behave as ahomogeneous effective medium. In this regard, the Backus average istypically considered as the low frequency, while ray theory defines thehigh frequency limit of the medium velocity. Embodiments utilize thisconcept, model anisotropic velocity in one scale and treat anisotropicvelocity as isotropic velocity in another scale.

Backus derived the effective elastic constants for a stratified mediumcomposed of TI layers in the long-wavelength limit. This method replacesthe stratified medium with an equivalent TI medium, and the fine layerscan be either isotropic or anisotropic with a spatially periodic ornon-periodic pattern. In the case of TI fine layers, the general elasticstiffness constants for the equivalent TI medium can be written asfollows using the Backus model:

C ₁₁ =

c ₁₁ −c ₁₃ ² c ₃₃ ⁻¹

+

c ₃₃ ⁻¹

⁽⁻¹⁾

c ₁₃ c ₃₃ ⁻¹

²

C ₁₂ =

c ₁₂ −c ₁₃ ² c ₃₃ ⁻¹

+

c ₃₃ ⁻¹

⁽⁻¹⁾

c ₁₃ c ₃₃ ⁻¹

²

C₁₃=

c₃₃ ⁻¹

⁽⁻¹⁾

c₁₃c₃₃ ⁻¹

C₃₃=

c₃₃ ⁻¹

⁽⁻¹⁾

C₄₄=

c₄₄ ⁻¹

⁽⁻¹⁾

C₆₆=

c₆₆

  (6)

The brackets indicate averages of the enclosed properties weighted bytheir volumetric proportions. C_(IJ) and c_(IJ) are referring to theconstants of elastic TI equivalent medium and fine layers. If individualfine layers are isotropic, the equivalent medium is still a TI medium.The elastic constants of such a TI medium can be calculated usingequation (6) as illustrated in FIG. 2, which illustrates the stiffnesstensor for isotropic and TI mediums. Backus showed that periodicrepetition of two isotropic layers each with two elastic constants cancreate a TI medium with five independent elastic constants. Asillustrated in FIG. 2, the brackets indicate averages of the enclosedproperties weighted by their volumetric proportions, and λ and μ areLame's parameters for each isotropic layer.

This infers that a TI medium stiffness matrix (C_(IJ) ) can be createdby adding some isotropic layers together and averaging their elasticproperties using the isotropic Backus model. The Backus model can besimplified even more considering normal wave propagation to the finelayering. It can be expressed as follows for a medium containing twolayers:

$\begin{matrix}{{\frac{1}{\rho \; V_{P}^{2}} = {\frac{f_{1}}{\rho_{1}\; V_{P\; 1}^{2}} + \frac{f_{2}}{\rho_{2}\; V_{P\; 2}^{2}}}}{\frac{1}{\rho \; V_{S}^{2}} = {\frac{f_{1}}{\rho_{1}\; V_{S\; 1}^{2}} + \frac{f_{2}}{\rho_{2}\; V_{S\; 2}^{2}}}}} & (7)\end{matrix}$

Here, V_(p) and V_(s) are the equivalent TI medium P- and S-wavevelocities, and ρ is their bulk density. The variables f₁ and f₂ are thevolume fractions of composed fine layers. The index numbers 1 and 2refer to these layers and their elastic parameters such as velocitiesand densities.

Equation (7) assumes that the two layers, e.g., the isotropic layer andthe effective anisotropic layer, are on a much finer scale than theseismic wavelengths. In this case, the waves will average the physicalproperties of the fine layers, so that the material becomes ahomogeneous effective medium with velocities for plane-wave propagationnormal to the layering.

In accordance with an embodiment for modelling elastic constants in a TImedium, the TI medium can be considered as a stack of two periodiclayers, either isotropic or anisotropic or a combination of isotropicand anisotropic. A model of the subsurface is created containing twolayers: a first layer containing all anisotropic factors and componentsand a second layer representing the balance of the subsurface medium andlacking any anisotropic behavior. All anisotropic minerals, e.g., clay,and factors, e.g., fracture and cracks, are considered separately in thefirst or anisotropic layer, and the rest of the subsurface medium withisotropic behavior is the second isotropic layer.

Based on the Backus model, the equivalent effective medium containingthese two layers (isotropic and effective anisotropic) behaves as a TImedium. Therefore, splitting a given subsurface medium into two separatelayers secures a TI behavior for the equivalent effective medium. Thecombination between the isotropic and effective anisotropic layersfacilitates manipulating the anisotropy degree for the equivalent mediumsuch that the lowest anisotropy is possible when both layers areisotropic (without any anisotropic factor included into the rock physicsmodeling) and highest anisotropy happens when rock physics modelling isdone and anisotropic factors are intensified in the anisotropic layer.

Referring now to FIG. 3, an embodiment of the workflow for modelling ananisotropic medium 300 is illustrated. As illustrated, the workflowincludes four steps. In a first step, a subsurface medium 302, which isassumed to be a TI subsurface medium, is divided into two layers, aneffective anisotropic layer 304 and an isotropic layer 306. A suite ofwell-log data has been acquired from at least one well passing throughthe subsurface medium parallel to the symmetry axis of the TI subsurfacemedium. This suite of well-log data contains elastic log data includingp-wave velocity, s-wave velocity and density. In the second step 308,rock physics modelling of the effective anisotropic layer 304, which isassumed as a TI medium is performed along the symmetry axis of the TImedium to generated elastic parameters for the effective anisotropiclayer. These elastic parameters include, but are not limited to, theeffective anisotropic layer p-wave velocity, the effective anisotropiclayer s-wave velocity and the effective anisotropic layer density. Inone embodiment, rock physics modelling combines all the initialcomponents and elements of the anisotropic layer using a rock physicsmodel that is suitable to the physical constituents of the subsurface.These initial components include, but are not limited to, texturalalignment constituents 310 such as grains and crystals, oriented cracksand micro-fractures 312 and mineralogy constituents 314 such as clays.In one embodiment, the volume of the anisotropic layer contains shale asthe main constituent.

The selection of the rock physics model to be used in generating theelastic parameters relies on information regarding the source ofanisotropy in the anisotropic layer. If the source of anisotropy isrelated to intrinsic factors such as subsurface medium microstructure,e.g., grain alignment, then a rock physics model like the Xu & White(1995) model is used to include these intrinsic factors, e.g., clayelastic properties, in the anisotropic layer. The Xu & White rockphysics model takes pore shape into account and can be considered formixing anisotropic minerals, e.g., clay with their pore space, i.e.,total porosity minus effective porosity, and pore fluid, i.e., claybound water. If the source of anisotropy is related to extrinsic factorssuch as cracks and fractures, then a rock physics model such as theHudson (1980) model is used to model the effective anisotropic layer asa cracked media. In one embodiment, it is assumed that all cracks with apreferential direction (source of TI medium) are located within theeffective anisotropic layer while the non-cracked portion of thesubsurface medium constitutes the isotropic layer.

Anisotropic minerals such as clay have a wide range of elasticproperties that differ from one type to another or even with depth inthe subsurface due to the digenesis. Even Hudson model utilizes someinformation on crack and fracture characteristics. In one embodiment,the effective anisotropic layer parameters are validated by couplingthis workflow to other data sources, for example, core ultrasonicmeasurement for different angles. Therefore, input mineral elasticproperties or crack properties are updated based on the core measurementdata taken at different incident angles. In addition, boundary modelssuch as the Hashin-Shetrikman boundary model (1963) or the Voigt (1890)and Reuss (1929) models can be used to define the boundaries foranisotropic changes in a given medium.

The output of the second step is an effective anisotropic medium thataverages all elastic properties and fractions of the initial componentswithin the anisotropic layer. It is assumed that the effectiveanisotropic layer contains all anisotropic information with C_(ij) asthe stiffness matrix. The anisotropic factors included in this layer areon a scale smaller than the wavelength, and this effective anisotropiclayer is seen as an effective isotropic layer with two effective Lame'sparameters to the seismic and sonic waves.

In the third step 316, the elastic properties of the isotropic layer arecalculated using the effective anisotropic layer elastic propertiescalculated in the second step 308 and the elastic log data acquired fromthe well-logs, i.e., V_(P), V_(S) and ρ, which can also be referred toas the measured elastic logs. The isotropic layer elastic propertiesinclude, but are not limited to, isotropic layer p-wave velocity,isotropic layer s-wave velocity and isotropic layer density. Therefore,in this equation all parameters are available to calculate the isotropiclayer elastic properties from the anisotropic layer elastic propertiesand the measured elastic properties from well logs. Therefore, onelayer, the complete TI subsurface medium, has been downscaled into twolayers, the isotropic layer and the effective anisotropic layer.

In one embodiment, equation (7), which is a simplified version of theBackus model for normal incidence, is used to calculate the isotropiclayer elastic properties. Equation (7) assumes a vertical well andhorizontally stratified layer. As illustrated, the modelled elasticproperties for the effective anisotropic layer are subtracted from theelastic log data properties to calculate the residual. This differenceor residual is the isotropic layer elastic properties. This stepdownscales measured elastic logs for a single TI layer into twoequivalent layers such that the summation of the two layers isequivalent to the one measured subsurface layer. The effectiveanisotropic layer is already modelled in step two 308, and allanisotropy information with a scale smaller than the wavelength isincluded during the modeling.

From steps 2 and 3, 308 and 316, the elastic parameters (Vp, Vs and ρ)of both layers were obtained, and in one embodiment these elasticparameters are converted into the elastic stiffness tensor members(c_(ij)) for each layer. For vertical wells and horizontal layers in theTI subsurface medium, any given well is vertical, perpendicular to theisotropic layers, parallel to the direction of the symmetry axis andaligned with the direction of the wave vector. Therefore, as discussedabove, the value of θ is set to 0 in equation (2), which simplifies thisequation into equation (7). Using equation (7), the elastic parametersfor the isotropic layer and the elastics parameters for the effectiveanisotropic layer are used to determine the Lame's parameters for eachlayer. For the isotropic layer, the independent elastic stiffness tensormembers are c₁₁ _(I) and c₁₂ _(I) , and for the effective anisotropiclayer, the independent elastic stiffness tensor members are c₁₁ _(A) andc₁₂ _(A) . Note that Backus model is valid for long-wavelength. Thismeans that each of these two layers are seen as isotropic layer by wavepropagation (anisotropic factors are in much smaller scale compared withwavelength as described herein). This infers that these two layers canbe expressed by using their Lame's parameters (λ and μ) in the sonicwavelength scale. Therefore, the calculated velocities in the symmetrydirection (calculated in the previous step) for each layer will betranslated into Lame's parameter for each layer.

The fourth step 318 uses equation (6) to upscale the two layers 304,306into the one layer 302 such that the effective elastic tensor within thesingle layer TI subsurface medium can be determined. In one embodiment,equation (6) is used to generate the effective elastic stiffness tensorsfor the upscaled single layer TI subsurface medium. The effectiveelastic stiffness tensor of these two layers using equation (6) is C₁₁,C₁₂, C₁₃, C₃₃ and C₄₄. In the long-wavelength as being assumed byBackus, each layer is seen as isotropic with two independent elasticstiffness members. In such cases, equation (6) can be written in termsof Lame's parameters as been given in FIG. 2.

Having established the elastic tensor members for each of the twolayers, a single set of five elastic tensor members is calculated forthe upscaled single layer TI subsurface medium. In one embodiment,equation (6) is used with the sum of the Lame's parameters of two layersinto a single set of five independent elastic tensor members, C₁₁, C₁₂,C₁₃, C₃₃ and C₄₄. The result is that the isotropic layer and theeffective anisotropic layer have upscaled into the single layer TIsubsurface medium, and the elastic stiffness constant for that singlelayer is known from the full version of the Backus model.

In one embodiment, in addition to the elastic stiffness constant for theresulting single layer TI subsurface medium, the p-wave and s-wavevelocities are also calculated for the TI subsurface medium. In oneembodiment, the five independent elastic stiffness tensor members of theTI subsurface, C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄, are used in equation (2) tocalculate the velocities. The measured elastic parameters from theacquired elastic logs were only valid for θ=0. However, using the fiveelastic stiffness tensor members in combination with equation (2),facilitates the calculation of subsurface velocities for any value of θ.Therefore, using the first elastic stiffness tensor members with anygiven angle yields the associated subsurface velocities for that givenangle or along the associated direction of wave propagation. Therefore,the anisotropic velocities have been calculated using only the measuredwell logs by downscaling and again upscaling the measured logs in avertical well, i.e., θ=0.

Regarding Lame's parameters, these parameters are a combination of theelastic tensor members, which is illustrated, for example, FIG. 1.Lame's parameters are used to describe an Isotropic layer having onlytwo independent elastic stiffness tensor members, c₁₁ and c₁₂.Therefore, in one embodiment, Lame's parameters are used for theeffective anisotropic layer (which is isotropic in larger scale) andisotropic layer instead of the elastic stiffness tensors members, andthe resulting equations are simplified.

The upscaled velocity logs will be identical to the measured velocitylogs as the same logs were used in generating the two fine layers in theprevious step, although the five independent constants will be differentfor different scenarios. These five elastic constants will change in away that the modelled logs will be equivalent to the measured log alongthe symmetry axis (θ=0). But the (anisotropic) information used in thesecond step will affect modeled velocities when the incident angle ofthe wave front is different from zero (θ≠0). Furthermore, equation (2)can be used to model velocities based on the modelled elastic constantsat different incidence angle (θ).

As illustrated, the rock physics workflow uses four steps to modelelastic parameters in a TI subsurface medium. These steps reconstructmeasured velocities from well logs (V_(P) and V_(S)) by downscaling andupscaling the TI subsurface medium using the Backus model along thesymmetry axis. The simplest scenario occurs when the source ofanisotropy in the TI subsurface medium can be considered from finelayering, FIG. 2. This simplest scenario assumes a stack of isotropiclayers for the equivalent TI medium but has a least chance of occurringin reality as other factors will cause a layer to behave like ananisotropic layer, e.g., porosity and saturation. However, the degree ofanisotropy increases with an increase in the factors involved inanisotropic behavior, e.g., fractures. In either cases rock physicsmodelling of a first or effective anisotropic layer is used. The mainassumption in this step is that the modelled layer builds an isotropicor a TI subsurface medium with the symmetry axis along the well boredirection.

The results of the previous scenario, i.e., a fine layering source ofanisotropy, are then used in the second step, and the final equivalentTI medium from the previous scenario provides the effective medium forthe first or effective anisotropic layer that is used in the secondstep. If another reason or source of the first layer anisotropy isexpected, then an appropriate effective medium theory is used thatincludes such anisotropy into these layer elastic properties. In oneembodiment, the accuracy of these modelled effective anisotropic elasticproperties is confirmed using additional data. The additional data forthe TI medium can be obtained from ultrasonic measurements on cores oreven azimuthal inversion. Ultrasonic core measurement is a goodcandidate where velocities at different incident wave angle areprovided. Therefore, the modelling parameters in the second step areupdated in accordance with the best fit of the final equivalent mediumvelocities and velocities coming from ultrasonic measurements atdifferent wave incident angles except zero (θ=0). However, without suchinformation it is only possible to define a range for changes in the TImedium elastic constants. This range can represent scenarios from low tohigh anisotropy, where the actual anisotropy can be located somewhere inbetween them.

Embodiments provide improved systems and methods for modeling elasticstiffness tensors in TI subsurface mediums using a rock physicsworkflow. The rock physics workflow provides a fast approach toestimating anisotropy affecting parameters using only conventional welllogs from wells passing through the TI subsurface medium. Differenteffective medium models are coupled with the Backus model to extractelastic constants by downscaling and upscaling the elastic properties.The normal incidence Backus model is used to downscale well logs intotwo layers, and then the full Backus model on the same layers is appliedto upscale them into the single layer TI subsurface medium. Thisapproach can be coupled easily with any other workflows to model VTImediums.

Referring now to FIG. 4, embodiments are directed to a method formodeling an elastic stiffness tensor in a transverse isotropicsubsurface medium 400. Well log data are acquired for at least one wellpassing through the transverse isotropic subsurface medium. In oneembodiment, well log data are acquired for a plurality of well passingthrough the transverse isotropic subsurface medium. The wells passthrough the transverse isotropic subsurface medium perpendicular to theisotropic layers in the subsurface. For horizontal layers, the wells arevertical. In one embodiment, the well log data are acquired in adirection parallel to a symmetry axis passing through layers in thetransverse isotropic subsurface medium. The well log data includes, butis not limited to p-wave velocity, s-wave velocity and density.

The transverse isotropic subsurface medium is divided into two layers,an effective anisotropic layer and an isotropic layer 404. The sum of aneffective anisotropic layer volume fraction and an isotropic layervolume fraction equals one. Elastics parameters for the effectiveanisotropic layer are modeled 406. Suitable anisotropic layer effectiveelastic parameters include, but are not limited to, an anisotropic layerdensity, an effective anisotropic layer p-wave velocity and an effectiveanisotropic layer s-wave velocity. To model the anisotropic layereffective elastic parameters, an anisotropic layer density is calculatedas a volume weighted average of all anisotropic component densities inthe anisotropic layer, e.g., density of the anisotropic componentmultiplied by the volume of that component in the anisotropic layer. Theeffective anisotropic layer p-wave velocity and the effectiveanisotropic layer s-wave velocity are then modeled along a symmetryaxis, i.e., θ=0, of the transverse isotropic subsurface medium using arock physics model selected in accordance with a source of anisotropy inthe anisotropic layer. In one embodiment, the rock physics model isbased on and includes intrinsic factors in the anisotropy layer, e.g.,microstructure of the transverse isotropic subsurface medium such asgrain alignment, or extrinsic factors, e.g., cracks and fractures, inthe anisotropy layer. Suitable rock physics models include, but are notlimited to, the Xu & White model, the Hudson model, theHashin-Shetrikman boundary model, the Voit model and the Reuss model. Inone embodiment, additional anisotropy data that are external to and inaddition to the well log data are used in modeling the effectiveanisotropic layer p-wave velocity and the effective anisotropic layers-wave velocity. Suitable additional anisotropy data include at leastone of core data, core ultrasonic measurements for a plurality of wavepropagation angles and seismic data.

Next, the isotropic layer elastic parameters are modeled using theanisotropic layer elastic parameters and the acquired well log data,i.e., the input data. Suitable isotropic layer elastic parametersinclude, but are not limited to, an isotropic layer density, anisotropic layer p-wave velocity and an isotropic layer s-wave velocity.In one embodiment, modeling the isotropic layer elastic parametersincludes using measured p-wave velocity and measured s-wave velocityfrom the acquired well log data and modeled effective anisotropic layerp-wave velocity and effective anisotropic layer s-wave velocity in asimplified Backus model for a two layer transverse isotropic medium andwave propagation normal to layering in the two layer isotropic medium tomodel the isotropic layer elastic parameters, i.e., equation (7).

The modeled effective anisotropic layer elastic parameters and themodeled isotropic layer elastic parameters are used to upscale theeffective anisotropic layer and the isotropic layer into the transverseisotropic subsurface medium comprising a single layer 410 and todetermine the elastic stiffness tensor for the transverse isotropicsubsurface medium 412.

In one embodiment, using the modeled effective anisotropic layer elasticparameters and the modeled isotropic layer elastic parameters to upscalethe anisotropic layer includes using the effective anisotropic layerelastic parameters to determine two effective Lame's parameters (λ_(A)and μ_(A)) and using the isotropic layer elastic parameters to determinetwo Lame's parameters (λ_(I) and μ_(I)). The two effective anisotropiclayer elastic tensor members and the two isotropic layer tensors arecombined to yield five independent transverse isotropic subsurfacemedium elastic tensor members (C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄), i.e.,equation (6), which is the full Backus model.

In one embodiment, upscaling the modeled isotropic layer elasticparameters and the effective anisotropic layer elastic parametersfurther includes using the anisotropic layer elastic parameters todetermine two effective anisotropic layer elastic tensor elements whichresults in two effective Lame's parameter for the anisotropic layer(λ_(A) and μ_(A)) and using the isotropic layer elastic parameters todetermine two Lame's parameters for the isotropic layer (λ_(I) andμ_(I)). The two effective anisotropic layer elastic tensor members andthe Lame's parameters are then combined to yield the five independenttransverse isotropic subsurface medium elastic tensor members of thesubsurface TI medium (C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄).

In one embodiment, the five transverse isotopic subsurface elastictensor members are used to calculate transverse isotropic subsurfacemedium p-wave velocities and transverse isotropic subsurface mediums-wave velocities for a plurality of wave propagation angles withrespect to an axis of symmetry in the transverse isotropic subsurfacemedium 414.

Referring now to FIG. 5, embodiments are directed to a computing system500 for modeling an elastic stiffness tensor in a transverse isotropicsubsurface medium. In one embodiment, a computing device for performingthe calculations as set forth in the above-described embodiments may beany type of computing device capable of obtaining, processing andcommunicating multi-vintage seismic data associated with seismic surveysconducted at different time periods. The computing system 500 includes acomputer or server 502 having one or more central processing units 504in communication with a communication module 506, one or moreinput/output devices 510 and at least one storage device 508.

The communication module is used to obtain well log data, core data anddipole sonic data for a plurality of wells passing through a subsurfaceregion in a project area. These well log data, core data and dipolesonic data can be obtained, for example, through the input/outputdevices. The well log data, core data and dipole sonic data are storedin the storage device. In addition, the storage device is used to storewell log data for at least one well passing through the transverseisotropic subsurface medium. The input/output device can also be used tocommunicate or display the elastic stiffness tensor, modeled p-wave ands-wave velocities and any images or models generated for the transverseisotropic subsurface medium, for example, to a user of the computingsystem.

The processer is in communication with the communication module andstorage device and is configured to divide the transverse isotropicsubsurface medium into an effective anisotropic layer and an isotropiclayer such that a sum of an anisotropic layer volume fraction and anisotropic layer volume fraction equals one, model effective anisotropiclayer elastic parameters, model isotropic layer elastic parameters usingthe effective anisotropic layer elastic parameters and the acquired welllog data and use the modeled effective anisotropic layer elasticparameters and the modeled isotropic layer elastic parameters to upscalethe effective anisotropic layer and the isotropic layer into thetransverse isotropic subsurface medium comprising a single layer and todetermine the elastic stiffness tensor for the transverse isotropicsubsurface medium.

Suitable embodiments for the various components of the computing systemare known to those of ordinary skill in the art, and this descriptionincludes all known and future variants of these types of devices. Thecommunication module provides for communication with other computingsystems, databases and data acquisition systems across one or more localor wide area networks 512. This includes both wired and wirelesscommunication. Suitable input/output devices include keyboards, pointand click type devices, audio devices, optical media devices and visualdisplays.

Suitable storage devices include magnetic media such as a hard diskdrive (HDD), solid state memory devices including flash drives, ROM andRAM and optical media. The storage device can contain data as well assoftware code for executing the functions of the computing system andthe functions in accordance with the methods described herein.Therefore, the computing system 500 can be used to implement the methodsdescribed above associated with predicting hydraulic fracture treatment.Hardware, firmware, software or a combination thereof may be used toperform the various steps and operations described herein.

Methods and systems in accordance with embodiments can be hardwareembodiments, software embodiments or a combination of hardware andsoftware embodiments. In one embodiment, the methods described hereinare implemented as software. Suitable software embodiments include, butare not limited to, firmware, resident software and microcode. Inaddition, methods and systems can take the form of a computer programproduct accessible from a computer-usable or computer-readable mediumproviding program code for use by or in connection with a computer,logical processing unit or any instruction execution system. In oneembodiment, a machine-readable or computer-readable medium contains amachine-executable or computer-executable code that when read by amachine or computer causes the machine or computer to perform a methodfor modeling an elastic stiffness tensor in a transverse isotropicsubsurface medium in accordance with embodiments and to thecomputer-executable code itself. The machine-readable orcomputer-readable code can be any type of code or language capable ofbeing read and executed by the machine or computer and can be expressedin any suitable language or syntax known and available in the artincluding machine languages, assembler languages, higher levellanguages, object oriented languages and scripting languages.

As used herein, a computer-usable or computer-readable medium can be anyapparatus that can contain, store, communicate, propagate, or transportthe program for use by or in connection with the instruction executionsystem, apparatus, or device. Suitable computer-usable or computerreadable mediums include, but are not limited to, electronic, magnetic,optical, electromagnetic, infrared, or semiconductor systems (orapparatuses or devices) or propagation mediums and includenon-transitory computer-readable mediums. Suitable computer-readablemediums include, but are not limited to, a semiconductor or solid statememory, magnetic tape, a removable computer diskette, a random accessmemory (RAM), a read-only memory (ROM), a rigid magnetic disk and anoptical disk. Suitable optical disks include, but are not limited to, acompact disk-read only memory (CD-ROM), a compact disk-read/write(CD-R/W) and DVD.

The disclosed embodiments provide a computing device, software andmethod for method for modeling an elastic stiffness tensor in atransverse isotropic subsurface medium. It should be understood thatthis description is not intended to limit the invention. On thecontrary, the embodiments are intended to cover alternatives,modifications and equivalents, which are included in the spirit andscope of the invention. Further, in the detailed description of theembodiments, numerous specific details are set forth in order to providea comprehensive understanding of the invention. However, one skilled inthe art would understand that various embodiments may be practicedwithout such specific details.

Although the features and elements of the present embodiments aredescribed in the embodiments in particular combinations, each feature orelement can be used alone without the other features and elements of theembodiments or in various combinations with or without other featuresand elements disclosed herein. The methods or flowcharts provided in thepresent application may be implemented in a computer program, software,or firmware tangibly embodied in a computer-readable storage medium forexecution by a geophysics dedicated computer or a processor.

This written description uses examples of the subject matter disclosedto enable any person skilled in the art to practice the same, includingmaking and using any devices or systems and performing any incorporatedmethods. The patentable scope of the subject matter is defined by theclaims, and may include other examples that occur to those skilled inthe art. Such other examples are intended to be within the scope of theclaims.

1. A method for modeling an elastic stiffness tensor in a transverseisotropic subsurface medium, the method comprising: acquiring well logdata for at least one well passing through the transverse isotropicsubsurface medium; dividing the transverse isotropic subsurface mediuminto an effective anisotropic layer and an isotropic layer such that asum of an effective anisotropic layer volume fraction and an isotropiclayer volume fraction equals one; modeling effective anisotropic layerelastic parameters; modeling isotropic layer elastic parameters usingthe anisotropic layer elastic parameters and the acquired well log data;using the modeled effective anisotropic layer elastic parameters and themodeled isotropic layer elastic parameters to upscale, with a processor,the effective anisotropic layer and the isotropic layer into thetransverse isotropic subsurface medium comprising a single layer and todetermine the elastic stiffness tensor for the transverse isotropicsubsurface medium; and calculating transverse isotropic subsurfacemedium p-wave velocities and transverse isotropic subsurface mediums-wave velocities based on the elastic stiffness tensor, wherein thewell log data is acquired inside a well and is indicative of physicalproperties of the subsurface medium.
 2. The method of claim 1, whereinacquiring the well log data comprises acquiring the well log data in adirection parallel to a symmetry axis passing through layers in thetransverse isotropic subsurface medium.
 3. The method of claim 1,wherein the effective anisotropic layer elastic parameters comprise ananisotropic layer density, an anisotropic layer p-wave velocity and ananisotropic layer s-wave velocity.
 4. The method of claim 1, wherein theisotropic layer elastic parameters comprise an isotropic layer density,an isotropic layer p-wave velocity and an isotropic layer s-wavevelocity.
 5. The method of claim 1, wherein modeling the effectiveanisotropic layer elastic parameters further comprises: calculating ananisotropic layer density as a volume weighted average of allanisotropic component densities in the anisotropic layer; and modelingan anisotropic layer p-wave velocity and an anisotropic layer s-wavevelocity along a symmetry axis of the transverse isotropic subsurfacemedium using a rock physics model selected in accordance with a sourceof anisotropy in the anisotropic layer.
 6. The method of claim 5,wherein the rock physics model comprises intrinsic factors in theanisotropy layer or extrinsic factors in the anisotropy layer.
 7. Themethod of claim 5, wherein modeling the effective anisotropic layerp-wave velocity and anisotropic layer s-wave velocity further comprisesusing additional anisotropy data comprising at least one of core data,core ultrasonic measurements for a plurality of wave propagation anglesand seismic data.
 8. The method of claim 1, wherein modeling theisotropic layer elastic parameters further comprises using measuredp-wave velocity and measured s-wave velocity from the acquired well logdata and modeled effective anisotropic layer p-wave velocity andeffective anisotropic layer s-wave velocity in a simplified Backus modelfor a two layer transverse isotropic medium and wave propagation normalto layering in the two layer isotropic medium to model the isotropiclayer elastic parameters.
 9. The method of claim 1, wherein using themodeled effective anisotropic layer elastic parameters and the modeledisotropic layer elastic parameters to upscale the effective anisotropiclayer and the isotropic layer into the transverse isotropic subsurfacemedium comprises: using the effective anisotropic layer elasticparameters to determine two effective Lame's parameters (λ_(A) andμ_(A)); and using the isotropic layer elastic parameters to determinetwo Lame's parameters (λ_(I) and μ_(I)).
 10. The method of claim 9,wherein using the modeled effective anisotropic layer elastic parametersand the modeled isotropic layer elastic parameters to upscale theeffective anisotropic layer and the isotropic layer into the transverseisotropic subsurface medium further comprises combining the effectiveanisotropic layer Lame's parameters (λ_(A) and μ_(A)) and the isotropiclayer Lame's parameters (μ_(I) and μ_(I)) to yield five independentmembers of the transverse isotropic subsurface medium elastic tensor(C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄).
 11. The method of claim 10, whereincombining the two effective Lame's parameters of the anisotropic layerwith two Lame's parameters of the isotropic layer further comprisesusing a full Backus model.
 12. The method of claim 10, furthercomprising using the five independent transverse isotopic subsurfaceelastic tensor members (C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄) to calculate thetransverse isotropic subsurface medium p-wave velocities and transverseisotropic subsurface medium s-wave velocities for a plurality of wavepropagation angles with respect to an axis of symmetry in the transverseisotropic subsurface medium.
 13. The method of claim 1, wherein usingthe modeled effective anisotropic layer elastic parameters and themodeled isotropic layer elastic parameters to upscale the effectiveanisotropic layer and the isotropic layer into the transverse isotropicsubsurface medium further comprises: using the effective anisotropiclayer elastic parameters to determine effective anisotropic layer Lame'sparameters (λ_(A) and μ_(A)); using the isotropic layer elasticparameters to determine two Lame's parameters for the isotropic layer(λ_(I) and μ_(I)); and combining the effective Lame's parameters for theanisotropic layer and the Lame's parameters for the isotropic layer toyield independent transverse isotropic subsurface medium elastic tensormembers for the subsurface medium.
 14. A computer-readablenon-transitory medium containing computer-executable code that when readby a computer causes the computer to perform a method for modeling anelastic stiffness tensor in a transverse isotropic subsurface medium,the method comprising: acquiring well log data for at least one wellpassing through the transverse isotropic subsurface medium; dividing thetransverse isotropic subsurface medium into an effective anisotropiclayer and an isotropic layer such that a sum of an effective anisotropiclayer volume fraction and an isotropic layer volume fraction equals one;modeling effective anisotropic layer elastic parameters; modelingisotropic layer elastic parameters using the anisotropic layer elasticparameters and the acquired well log data; using the modeled effectiveanisotropic layer elastic parameters and the modeled isotropic layerelastic parameters to upscale the effective anisotropic layer and theisotropic layer into the transverse isotropic subsurface mediumcomprising a single layer and to determine the elastic stiffness tensorfor the transverse isotropic subsurface medium; and calculatingtransverse isotropic subsurface medium p-wave velocities and transverseisotropic subsurface medium s-wave velocities based on the elasticstiffness tensor, wherein the well log data is acquired inside a welland is indicative of physical properties of the subsurface medium. 15.The computer-readable medium of claim 14, wherein modeling the effectiveanisotropic layer elastic parameters further comprises: calculating ananisotropic layer density as a volume weighted average of allanisotropic component densities in the anisotropic layer; and modelingan effective anisotropic layer p-wave velocity and an effectiveanisotropic layer s-wave velocity along a symmetry axis of thetransverse isotropic subsurface medium using a rock physics modelselected in accordance with a source of anisotropy in the anisotropiclayer.
 16. The computer-readable medium of claim 14, wherein modelingthe isotropic layer elastic parameters further comprises using measuredp-wave velocity and measured s-wave velocity from the acquired well logdata and modeled effective anisotropic layer p-wave velocity andanisotropic layer s-wave velocity in a simplified Backus model for a twolayer transverse isotropic medium and wave propagation normal tolayering in the two layer isotropic medium to model the isotropic layerelastic parameters.
 17. The computer-readable of claim 14, wherein usingthe modeled effective anisotropic layer elastic parameters and themodeled isotropic layer elastic parameters to upscale the effectiveanisotropic layer and the isotropic layer into the transverse isotropicsubsurface medium further comprises: using the effective anisotropiclayer elastic parameters to determine two effective Lame's parameters(λ_(A) and μ_(A)); and using the isotropic layer elastic parameters todetermine two Lame's parameters (λ_(I) and μ_(I)).
 18. Thecomputer-readable medium of claim 17, wherein using the modeledeffective anisotropic layer elastic parameters and the modeled isotropiclayer elastic parameters to upscale the effective anisotropic layer andthe isotropic layer into the transverse isotropic subsurface mediumfurther comprises: combining the effective Lame's parameters for theanisotropic layer and the Lame's parameters for the isotropic layer toyield independent elastic tensor members for the transverse isotropicsubsurface medium.
 19. The computer-readable medium of claim 18, whereincombining the two effective Lame's parameters for the anisotropic layerand the Lame's parameters for the isotropic layer further comprisesusing a full Backus model.
 20. A computing system for modeling anelastic stiffness tensor in a transverse isotropic subsurface medium,the computing system comprising: a storage device comprising well logdata for at least one well passing through the transverse isotropicsubsurface medium; and a processor in communication with the storagedevice and configured to: divide the transverse isotropic subsurfacemedium into an effective anisotropic layer and an isotropic layer suchthat a sum of an effective anisotropic layer volume fraction and anisotropic layer volume fraction equals one; model effective anisotropiclayer elastic parameters; model isotropic layer elastic parameters usingthe effective anisotropic layer elastic parameters and the acquired welllog data; use the modeled effective anisotropic layer elastic parametersand the modeled isotropic layer elastic parameters to upscale theeffective anisotropic layer and the isotropic layer into the transverseisotropic subsurface medium comprising a single layer and to determineindependent members of the elastic stiffness tensor for the transverseisotropic subsurface medium; and calculate transverse isotropicsubsurface medium p-wave velocities and transverse isotropic subsurfacemedium s-wave velocities based on the elastic stiffness tensor, whereinthe well log data is acquired inside a well and is indicative ofpetrophysical properties of the subsurface medium.